#9938079

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4109 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2,7,9)(3,8,4,14)(5,10,15,13)(6,12,16,11)(17,18,22,25)(19,21,27,23)(20,24,29,32)(26,30,31,28)(34,39)(35,42)(37,45)(44,47);(1,3)(2,8)(4,7)(5,12)(6,13)(9,14)(10,16)(11,15)(17,19,22,27)(18,23,25,21)(20,26,29,31)(24,28,32,30)(33,34,38,39)(35,36,42,40)(37,41,45,46)(43,44,48,47);(2,9)(3,4)(10,13)(11,12)(17,19,22,27)(18,23,25,21)(20,26,29,31)(24,28,32,30)(33,35,38,42)(34,40,39,36)(37,43,45,48)(41,47,46,44);(1,4,7,3)(2,8,9,14)(5,11,15,12)(6,10,16,13)(17,19)(18,21)(20,26)(22,27)(23,25)(24,30)(28,32)(29,31)(33,35)(34,36)(37,43)(38,42)(39,40)(41,44)(45,48)(46,47);(1,5)(2,10)(3,12)(4,11)(6,14)(7,15)(8,16)(9,13)(17,20)(18,24)(19,26)(21,30)(22,29)(23,28)(25,32)(27,31)(33,36)(34,35)(37,44)(38,40)(39,42)(41,43)(45,47)(46,48);(1,6,7,16)(2,11,9,12)(3,13,4,10)(5,8,15,14)(17,21)(18,19)(20,28)(22,23)(24,31)(25,27)(26,32)(29,30)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=g,d^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9845275);